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Shared Qs (031)


  1. Question

    Integration reverses differentiation. We will practice “undoing” the power rule by finding an antiderivative.

    Let \(f(x)\) be an unknown polynomial. Its derivative is known.

    \[f'(x)={-15}x^{4}+{16}x^{3}-{18}x^{2}\]

    To undo the power rule, each term’s power is increased by 1, and then the coefficient is divided by the new power. This gets us most of the way.

    A (slightly annoying) complication occurs because adding any constant to \(f(x)\) would not change its derivative, since the derivative of a constant is 0. This means there are infinitely many possible antiderivatives with a constant of integration equaling any real number. If we know a single solution to \(y=f(x)\) (a single point on the curve), then the constant of integration can be determined. In this case, let’s enforce the following condition:

    \[f(2)=-84\]

    We know the integral can be expressed as a polynomial with 4 terms, where \(j>k>l\):

    \[f(x)=Jx^{j}+Kx^{k}+Lx^l+C\]

    Find the parameters.



    Solution


  2. Question

    Integration reverses differentiation. We will practice “undoing” the power rule by finding an antiderivative.

    Let \(f(x)\) be an unknown polynomial. Its derivative is known.

    \[f'(x)={-10}x^{4}-{32}x^{3}-{4}{}\]

    To undo the power rule, each term’s power is increased by 1, and then the coefficient is divided by the new power. This gets us most of the way.

    A (slightly annoying) complication occurs because adding any constant to \(f(x)\) would not change its derivative, since the derivative of a constant is 0. This means there are infinitely many possible antiderivatives with a constant of integration equaling any real number. If we know a single solution to \(y=f(x)\) (a single point on the curve), then the constant of integration can be determined. In this case, let’s enforce the following condition:

    \[f(-4)=11\]

    We know the integral can be expressed as a polynomial with 4 terms, where \(j>k>l\):

    \[f(x)=Jx^{j}+Kx^{k}+Lx^l+C\]

    Find the parameters.



    Solution


  3. Question

    Integration reverses differentiation. We will practice “undoing” the power rule by finding an antiderivative.

    Let \(f(x)\) be an unknown polynomial. Its derivative is known.

    \[f'(x)={-12}x^{3}+{24}x^{2}-{18}x^{}\]

    To undo the power rule, each term’s power is increased by 1, and then the coefficient is divided by the new power. This gets us most of the way.

    A (slightly annoying) complication occurs because adding any constant to \(f(x)\) would not change its derivative, since the derivative of a constant is 0. This means there are infinitely many possible antiderivatives with a constant of integration equaling any real number. If we know a single solution to \(y=f(x)\) (a single point on the curve), then the constant of integration can be determined. In this case, let’s enforce the following condition:

    \[f(2)=-26\]

    We know the integral can be expressed as a polynomial with 4 terms, where \(j>k>l\):

    \[f(x)=Jx^{j}+Kx^{k}+Lx^l+C\]

    Find the parameters.



    Solution


  4. Question

    Integration reverses differentiation. We will practice “undoing” the power rule by finding an antiderivative.

    Let \(f(x)\) be an unknown polynomial. Its derivative is known.

    \[f'(x)={-35}x^{6}-{36}x^{5}+{35}x^{4}\]

    To undo the power rule, each term’s power is increased by 1, and then the coefficient is divided by the new power. This gets us most of the way.

    A (slightly annoying) complication occurs because adding any constant to \(f(x)\) would not change its derivative, since the derivative of a constant is 0. This means there are infinitely many possible antiderivatives with a constant of integration equaling any real number. If we know a single solution to \(y=f(x)\) (a single point on the curve), then the constant of integration can be determined. In this case, let’s enforce the following condition:

    \[f(-2)=39\]

    We know the integral can be expressed as a polynomial with 4 terms, where \(j>k>l\):

    \[f(x)=Jx^{j}+Kx^{k}+Lx^l+C\]

    Find the parameters.



    Solution


  5. Question

    Integration reverses differentiation. We will practice “undoing” the power rule by finding an antiderivative.

    Let \(f(x)\) be an unknown polynomial. Its derivative is known.

    \[f'(x)={20}x^{4}-{12}x^{3}-{27}x^{2}\]

    To undo the power rule, each term’s power is increased by 1, and then the coefficient is divided by the new power. This gets us most of the way.

    A (slightly annoying) complication occurs because adding any constant to \(f(x)\) would not change its derivative, since the derivative of a constant is 0. This means there are infinitely many possible antiderivatives with a constant of integration equaling any real number. If we know a single solution to \(y=f(x)\) (a single point on the curve), then the constant of integration can be determined. In this case, let’s enforce the following condition:

    \[f(2)=2\]

    We know the integral can be expressed as a polynomial with 4 terms, where \(j>k>l\):

    \[f(x)=Jx^{j}+Kx^{k}+Lx^l+C\]

    Find the parameters.



    Solution


  6. Question

    A definite integral can be used to express the area under a curve. For example, say we wanted to find the area under \(f(x)=x^{}-x^{7}\) between \(x=0.43\) and \(x=0.81\). Let’s draw a picture.

    plot of chunk unnamed-chunk-2

    The area equals the definite integral of the function on interval [0.43, 0.81].

    \[A ~=~ \int_{0.43}^{0.81} \left(x^{}-x^{7} \right)~dx\]

    This definite integral can be evaluated from an antiderivative of \(f(x)\). Let \(g(x)\) be an antiderivative of \(f(x)\) such that \(g'(x)=f(x)\).

    \[A~=~g(0.81)-g(0.43)\]

    Find the area. The tolerance is \(\pm 0.001\).


    Solution


  7. Question

    A definite integral can be used to express the area under a curve. For example, say we wanted to find the area under \(f(x)=x^{3}-x^{9}\) between \(x=0.51\) and \(x=0.8\). Let’s draw a picture.

    plot of chunk unnamed-chunk-2

    The area equals the definite integral of the function on interval [0.51, 0.8].

    \[A ~=~ \int_{0.51}^{0.8} \left(x^{3}-x^{9} \right)~dx\]

    This definite integral can be evaluated from an antiderivative of \(f(x)\). Let \(g(x)\) be an antiderivative of \(f(x)\) such that \(g'(x)=f(x)\).

    \[A~=~g(0.8)-g(0.51)\]

    Find the area. The tolerance is \(\pm 0.001\).


    Solution


  8. Question

    A definite integral can be used to express the area under a curve. For example, say we wanted to find the area under \(f(x)=x^{}-x^{6}\) between \(x=0.49\) and \(x=0.87\). Let’s draw a picture.

    plot of chunk unnamed-chunk-2

    The area equals the definite integral of the function on interval [0.49, 0.87].

    \[A ~=~ \int_{0.49}^{0.87} \left(x^{}-x^{6} \right)~dx\]

    This definite integral can be evaluated from an antiderivative of \(f(x)\). Let \(g(x)\) be an antiderivative of \(f(x)\) such that \(g'(x)=f(x)\).

    \[A~=~g(0.87)-g(0.49)\]

    Find the area. The tolerance is \(\pm 0.001\).


    Solution


  9. Question

    A definite integral can be used to express the area under a curve. For example, say we wanted to find the area under \(f(x)=x^{}-x^{5}\) between \(x=0.54\) and \(x=0.8\). Let’s draw a picture.

    plot of chunk unnamed-chunk-2

    The area equals the definite integral of the function on interval [0.54, 0.8].

    \[A ~=~ \int_{0.54}^{0.8} \left(x^{}-x^{5} \right)~dx\]

    This definite integral can be evaluated from an antiderivative of \(f(x)\). Let \(g(x)\) be an antiderivative of \(f(x)\) such that \(g'(x)=f(x)\).

    \[A~=~g(0.8)-g(0.54)\]

    Find the area. The tolerance is \(\pm 0.001\).


    Solution


  10. Question

    A definite integral can be used to express the area under a curve. For example, say we wanted to find the area under \(f(x)=x^{3}-x^{7}\) between \(x=0.56\) and \(x=0.85\). Let’s draw a picture.

    plot of chunk unnamed-chunk-2

    The area equals the definite integral of the function on interval [0.56, 0.85].

    \[A ~=~ \int_{0.56}^{0.85} \left(x^{3}-x^{7} \right)~dx\]

    This definite integral can be evaluated from an antiderivative of \(f(x)\). Let \(g(x)\) be an antiderivative of \(f(x)\) such that \(g'(x)=f(x)\).

    \[A~=~g(0.85)-g(0.56)\]

    Find the area. The tolerance is \(\pm 0.001\).


    Solution


  11. Question

    A particle moves up and down along a 1-dimensional path. It’s acceleration (\(a(t)\) in \(\mathrm{\frac{m}{s^2}}\)) is a function of time (\(t\) in seconds).

    \[a(t)=-48t+12\]

    The particle’s velocity is also a function of time. Also, \(v(0)=4\).

    \[v(t) ~=~ 4 + \int_{0}^{t}a(t)\,dt\]

    The particle’s position is also a function of time. Also, \(x(0)=4\).

    \[x(t) ~=~ 4 + \int_{0}^{t}v(t)\,dt\]

    Evaluate position, velocity, and acceleration at \(t=0\), \(t=1\), and \(t=2\).

    t y(t) v(t) a(t)
    0
    1
    2


    Solution


  12. Question

    A particle moves up and down along a 1-dimensional path. It’s acceleration (\(a(t)\) in \(\mathrm{\frac{m}{s^2}}\)) is a function of time (\(t\) in seconds).

    \[a(t)=-36t+10\]

    The particle’s velocity is also a function of time. Also, \(v(0)=8\).

    \[v(t) ~=~ 8 + \int_{0}^{t}a(t)\,dt\]

    The particle’s position is also a function of time. Also, \(x(0)=2\).

    \[x(t) ~=~ 2 + \int_{0}^{t}v(t)\,dt\]

    Evaluate position, velocity, and acceleration at \(t=0\), \(t=1\), and \(t=2\).

    t y(t) v(t) a(t)
    0
    1
    2


    Solution


  13. Question

    A particle moves up and down along a 1-dimensional path. It’s acceleration (\(a(t)\) in \(\mathrm{\frac{m}{s^2}}\)) is a function of time (\(t\) in seconds).

    \[a(t)=-48t+20\]

    The particle’s velocity is also a function of time. Also, \(v(0)=6\).

    \[v(t) ~=~ 6 + \int_{0}^{t}a(t)\,dt\]

    The particle’s position is also a function of time. Also, \(x(0)=-1\).

    \[x(t) ~=~ -1 + \int_{0}^{t}v(t)\,dt\]

    Evaluate position, velocity, and acceleration at \(t=0\), \(t=1\), and \(t=2\).

    t y(t) v(t) a(t)
    0
    1
    2


    Solution


  14. Question

    A particle moves up and down along a 1-dimensional path. It’s acceleration (\(a(t)\) in \(\mathrm{\frac{m}{s^2}}\)) is a function of time (\(t\) in seconds).

    \[a(t)=-36t+4\]

    The particle’s velocity is also a function of time. Also, \(v(0)=8\).

    \[v(t) ~=~ 8 + \int_{0}^{t}a(t)\,dt\]

    The particle’s position is also a function of time. Also, \(x(0)=10\).

    \[x(t) ~=~ 10 + \int_{0}^{t}v(t)\,dt\]

    Evaluate position, velocity, and acceleration at \(t=0\), \(t=1\), and \(t=2\).

    t y(t) v(t) a(t)
    0
    1
    2


    Solution


  15. Question

    A particle moves up and down along a 1-dimensional path. It’s acceleration (\(a(t)\) in \(\mathrm{\frac{m}{s^2}}\)) is a function of time (\(t\) in seconds).

    \[a(t)=-54t+14\]

    The particle’s velocity is also a function of time. Also, \(v(0)=9\).

    \[v(t) ~=~ 9 + \int_{0}^{t}a(t)\,dt\]

    The particle’s position is also a function of time. Also, \(x(0)=2\).

    \[x(t) ~=~ 2 + \int_{0}^{t}v(t)\,dt\]

    Evaluate position, velocity, and acceleration at \(t=0\), \(t=1\), and \(t=2\).

    t y(t) v(t) a(t)
    0
    1
    2


    Solution


  16. Question

    A 3D shape’s base is the 2D region enclosed by \(y=0\) and \(y=f(x)=x^{0.8}-x^{5.5}\) on the \(xy\) plane. For every plane perpendicular to the \(x\) axis, with \(0<x<1\), there is a square cross section.

    I have attempted to draw this below.

    plot of chunk unnamed-chunk-2

    And here is a spinning animation. Does that help?

    plot of chunk unnamed-chunk-3

    Find the volume of the shape; the tolerance is \(\pm 0.001\) cubic units. You might find this video helpful. The general topic is “volume from cross sections”; you might try a variety of resources by searching this phrase in your favorite search engine.


    Solution


  17. Question

    A 3D shape’s base is the 2D region enclosed by \(y=0\) and \(y=f(x)=x^{0.5}-x^{2.4}\) on the \(xy\) plane. For every plane perpendicular to the \(x\) axis, with \(0<x<1\), there is a square cross section.

    I have attempted to draw this below.

    plot of chunk unnamed-chunk-2

    And here is a spinning animation. Does that help?

    plot of chunk unnamed-chunk-3

    Find the volume of the shape; the tolerance is \(\pm 0.001\) cubic units. You might find this video helpful. The general topic is “volume from cross sections”; you might try a variety of resources by searching this phrase in your favorite search engine.


    Solution


  18. Question

    A 3D shape’s base is the 2D region enclosed by \(y=0\) and \(y=f(x)=x^{0.7}-x^{8.6}\) on the \(xy\) plane. For every plane perpendicular to the \(x\) axis, with \(0<x<1\), there is a square cross section.

    I have attempted to draw this below.

    plot of chunk unnamed-chunk-2

    And here is a spinning animation. Does that help?

    plot of chunk unnamed-chunk-3

    Find the volume of the shape; the tolerance is \(\pm 0.001\) cubic units. You might find this video helpful. The general topic is “volume from cross sections”; you might try a variety of resources by searching this phrase in your favorite search engine.


    Solution


  19. Question

    A 3D shape’s base is the 2D region enclosed by \(y=0\) and \(y=f(x)=x^{1.7}-x^{9.3}\) on the \(xy\) plane. For every plane perpendicular to the \(x\) axis, with \(0<x<1\), there is a square cross section.

    I have attempted to draw this below.

    plot of chunk unnamed-chunk-2

    And here is a spinning animation. Does that help?

    plot of chunk unnamed-chunk-3

    Find the volume of the shape; the tolerance is \(\pm 0.001\) cubic units. You might find this video helpful. The general topic is “volume from cross sections”; you might try a variety of resources by searching this phrase in your favorite search engine.


    Solution


  20. Question

    A 3D shape’s base is the 2D region enclosed by \(y=0\) and \(y=f(x)=x^{1.1}-x^{3.6}\) on the \(xy\) plane. For every plane perpendicular to the \(x\) axis, with \(0<x<1\), there is a square cross section.

    I have attempted to draw this below.

    plot of chunk unnamed-chunk-2

    And here is a spinning animation. Does that help?

    plot of chunk unnamed-chunk-3

    Find the volume of the shape; the tolerance is \(\pm 0.001\) cubic units. You might find this video helpful. The general topic is “volume from cross sections”; you might try a variety of resources by searching this phrase in your favorite search engine.


    Solution


  21. Question

    A 3D shape is produced with elliptical cross sections. As \(x\) progresses from along the interval \([0,1]\), the cross section perpendicular to the \(x\) axis will be an ellipse with axes (and radii) parallel to the \(y\) and \(z\) axes. The (maximum and minimum) radii are power functions with respect to \(x\).

    \[r_1(x) ~=~ x^{1.45}\] \[r_2(x) ~=~ x^{0.38}\]

    plot of chunk unnamed-chunk-2

    plot of chunk unnamed-chunk-3

    Find the volume of the shape. The tolerance is \(\pm 0.01\) cubic units.

    As a hint, I’ll remind you that the area of an ellipse is found by multiplying the radii by each other and pi: \(A=\pi\cdot r_1 \cdot r_2\).


    Solution


  22. Question

    A 3D shape is produced with elliptical cross sections. As \(x\) progresses from along the interval \([0,1]\), the cross section perpendicular to the \(x\) axis will be an ellipse with axes (and radii) parallel to the \(y\) and \(z\) axes. The (maximum and minimum) radii are power functions with respect to \(x\).

    \[r_1(x) ~=~ x^{1.03}\] \[r_2(x) ~=~ x^{2.52}\]

    plot of chunk unnamed-chunk-2

    plot of chunk unnamed-chunk-3

    Find the volume of the shape. The tolerance is \(\pm 0.01\) cubic units.

    As a hint, I’ll remind you that the area of an ellipse is found by multiplying the radii by each other and pi: \(A=\pi\cdot r_1 \cdot r_2\).


    Solution


  23. Question

    A 3D shape is produced with elliptical cross sections. As \(x\) progresses from along the interval \([0,1]\), the cross section perpendicular to the \(x\) axis will be an ellipse with axes (and radii) parallel to the \(y\) and \(z\) axes. The (maximum and minimum) radii are power functions with respect to \(x\).

    \[r_1(x) ~=~ x^{0.67}\] \[r_2(x) ~=~ x^{1}\]

    plot of chunk unnamed-chunk-2

    plot of chunk unnamed-chunk-3

    Find the volume of the shape. The tolerance is \(\pm 0.01\) cubic units.

    As a hint, I’ll remind you that the area of an ellipse is found by multiplying the radii by each other and pi: \(A=\pi\cdot r_1 \cdot r_2\).


    Solution


  24. Question

    A 3D shape is produced with elliptical cross sections. As \(x\) progresses from along the interval \([0,1]\), the cross section perpendicular to the \(x\) axis will be an ellipse with axes (and radii) parallel to the \(y\) and \(z\) axes. The (maximum and minimum) radii are power functions with respect to \(x\).

    \[r_1(x) ~=~ x^{2.11}\] \[r_2(x) ~=~ x^{1.58}\]

    plot of chunk unnamed-chunk-2

    plot of chunk unnamed-chunk-3

    Find the volume of the shape. The tolerance is \(\pm 0.01\) cubic units.

    As a hint, I’ll remind you that the area of an ellipse is found by multiplying the radii by each other and pi: \(A=\pi\cdot r_1 \cdot r_2\).


    Solution


  25. Question

    A 3D shape is produced with elliptical cross sections. As \(x\) progresses from along the interval \([0,1]\), the cross section perpendicular to the \(x\) axis will be an ellipse with axes (and radii) parallel to the \(y\) and \(z\) axes. The (maximum and minimum) radii are power functions with respect to \(x\).

    \[r_1(x) ~=~ x^{1.86}\] \[r_2(x) ~=~ x^{1.64}\]

    plot of chunk unnamed-chunk-2

    plot of chunk unnamed-chunk-3

    Find the volume of the shape. The tolerance is \(\pm 0.01\) cubic units.

    As a hint, I’ll remind you that the area of an ellipse is found by multiplying the radii by each other and pi: \(A=\pi\cdot r_1 \cdot r_2\).


    Solution


  26. Question

    A bowl is designed as the revolution of the region between 3 curves: \[z=4x-44\]

    \[z=\frac{4}{144}x^2\]

    \[z=0\]

    The revolution occurs around the \(z\) axis. The bowl will be made of wood.

    plot of chunk unnamed-chunk-2

    Notice the point of intersection is \((12, 4)\). Below is a wireframe animation of the bowl.

    plot of chunk unnamed-chunk-3

    Find the volume of the wood composing the bowl. I’d recommend using the washer method. The tolerance is \(\pm 1\) cubic units.


    Solution


  27. Question

    A bowl is designed as the revolution of the region between 3 curves: \[z=4x-30\]

    \[z=\frac{6}{81}x^2\]

    \[z=0\]

    The revolution occurs around the \(z\) axis. The bowl will be made of wood.

    plot of chunk unnamed-chunk-2

    Notice the point of intersection is \((9, 6)\). Below is a wireframe animation of the bowl.

    plot of chunk unnamed-chunk-3

    Find the volume of the wood composing the bowl. I’d recommend using the washer method. The tolerance is \(\pm 1\) cubic units.


    Solution


  28. Question

    A bowl is designed as the revolution of the region between 3 curves: \[z=3x-25\]

    \[z=\frac{8}{121}x^2\]

    \[z=0\]

    The revolution occurs around the \(z\) axis. The bowl will be made of wood.

    plot of chunk unnamed-chunk-2

    Notice the point of intersection is \((11, 8)\). Below is a wireframe animation of the bowl.

    plot of chunk unnamed-chunk-3

    Find the volume of the wood composing the bowl. I’d recommend using the washer method. The tolerance is \(\pm 1\) cubic units.


    Solution


  29. Question

    A bowl is designed as the revolution of the region between 3 curves: \[z=2x-11\]

    \[z=\frac{9}{100}x^2\]

    \[z=0\]

    The revolution occurs around the \(z\) axis. The bowl will be made of wood.

    plot of chunk unnamed-chunk-2

    Notice the point of intersection is \((10, 9)\). Below is a wireframe animation of the bowl.

    plot of chunk unnamed-chunk-3

    Find the volume of the wood composing the bowl. I’d recommend using the washer method. The tolerance is \(\pm 1\) cubic units.


    Solution


  30. Question

    A bowl is designed as the revolution of the region between 3 curves: \[z=5x-61\]

    \[z=\frac{4}{169}x^2\]

    \[z=0\]

    The revolution occurs around the \(z\) axis. The bowl will be made of wood.

    plot of chunk unnamed-chunk-2

    Notice the point of intersection is \((13, 4)\). Below is a wireframe animation of the bowl.

    plot of chunk unnamed-chunk-3

    Find the volume of the wood composing the bowl. I’d recommend using the washer method. The tolerance is \(\pm 1\) cubic units.


    Solution


  31. Question

    An isosceles trapezoid is the region between the following curves:

    \[z ~=~ 0.65x\] \[z ~=~ -0.65x\] \[x ~=~ 2.16\] \[x ~=~ 9.38\]

    plot of chunk unnamed-chunk-2

    That trapezoid is revolved around the \(z\) axis to produce a toroid. A wireframe animation of the toroid is shown below.

    plot of chunk unnamed-chunk-3

    Find the volume of the toroid using shell integration. The tolerance is \(\pm 1\) cubic unit.


    Solution


  32. Question

    An isosceles trapezoid is the region between the following curves:

    \[z ~=~ 0.79x\] \[z ~=~ -0.79x\] \[x ~=~ 2.56\] \[x ~=~ 7.07\]

    plot of chunk unnamed-chunk-2

    That trapezoid is revolved around the \(z\) axis to produce a toroid. A wireframe animation of the toroid is shown below.

    plot of chunk unnamed-chunk-3

    Find the volume of the toroid using shell integration. The tolerance is \(\pm 1\) cubic unit.


    Solution


  33. Question

    An isosceles trapezoid is the region between the following curves:

    \[z ~=~ 0.95x\] \[z ~=~ -0.95x\] \[x ~=~ 2.2\] \[x ~=~ 8.74\]

    plot of chunk unnamed-chunk-2

    That trapezoid is revolved around the \(z\) axis to produce a toroid. A wireframe animation of the toroid is shown below.

    plot of chunk unnamed-chunk-3

    Find the volume of the toroid using shell integration. The tolerance is \(\pm 1\) cubic unit.


    Solution


  34. Question

    An isosceles trapezoid is the region between the following curves:

    \[z ~=~ 0.58x\] \[z ~=~ -0.58x\] \[x ~=~ 6.01\] \[x ~=~ 9.16\]

    plot of chunk unnamed-chunk-2

    That trapezoid is revolved around the \(z\) axis to produce a toroid. A wireframe animation of the toroid is shown below.

    plot of chunk unnamed-chunk-3

    Find the volume of the toroid using shell integration. The tolerance is \(\pm 1\) cubic unit.


    Solution


  35. Question

    An isosceles trapezoid is the region between the following curves:

    \[z ~=~ 0.41x\] \[z ~=~ -0.41x\] \[x ~=~ 2.48\] \[x ~=~ 6.08\]

    plot of chunk unnamed-chunk-2

    That trapezoid is revolved around the \(z\) axis to produce a toroid. A wireframe animation of the toroid is shown below.

    plot of chunk unnamed-chunk-3

    Find the volume of the toroid using shell integration. The tolerance is \(\pm 1\) cubic unit.


    Solution